Polynomial Matrices, Splitting Subspaces and Krylov Subspaces over Finite Fields

Abstract

Let T be a linear operator on an Fq-vector space V of dimension n. For any divisor m of n, an m-dimensional subspacewhere d = n/m. Let σ(m, d; T) denote the number of m-dimensional Tsplitting subspaces. Determining σ(m, d; T ) for an arbitrary operator T is an open problem. This problem is closely related to another open problem on Krylov spaces. We discuss this connection and give explicit formulae for σ(m, d; T ) in the case where the invariant factors of T satisfy certain degree conditions. A connection with another enumeration problem on polynomial matrices is also discussed.

Splitting subspaces and a finite field interpretation of the Touchard-Riordan Formula

“…In the running example, the opening nodes are 1, 2, 3, 7 and the closing nodes are 4, 5, 6, 8. A crossing of the chord diagram is a pair of arcs (i, j), (k, l) such that i < k < j < l. The chord diagram above has two crossings, namely (1,4), (2,6) and (1,4), (3,5). Given a fixed-point-free involution σ, let v(σ) denote the number of crossings of its chord diagram.…”

“…The number of T -splitting subspaces is known when T has an irreducible characteristic polynomial [3,5,8], is regular nilpotent [2], is regular split semisimple [18,19], or when the invariant factors satisfy certain degree constraints [1]. In this article, we consider the case where d = 2.…”